And I get most of it fine, except stuff like this:
(-10)^2 = 100, or (-2)^4 = 16

Because -10^2 is = -(10x10) isn't it? Which is -100, so I'm a little confused how it isn't a negative when the subtraction gets the power?

-(10*10) does equal -100 and does equal -10^2; however, (-10)^2 is not the same value - the parenthesis have actual meaning. Because terms inside parenthesis have higher operator precedence than the exponent operator, the negative sign stays with the ten and you have -10*-10, which equals 100.